Final answer:
The differential equation (d-1)²y = 6e⁻ᴇ is solved by determining the characteristic equation and its roots, leading to the general solution involving repeated roots. The answer is D) y = 6e⁻ᴇ + C1e⁻ᴇ + C2xe⁻ᴇ.
Step-by-step explanation:
To solve the differential equation (d-1)2y = 6ex, we need to recognize that this equation implies repeated linear operators acting on the function y. The repeated operator (d - 1) suggests that we are dealing with repeated roots of the characteristic equation associated with a linear differential equation with constant coefficients. Let's proceed step by step.
First, we identify the characteristic equation corresponding to (d - 1)2, which is (r - 1)2 = 0. The roots are r = 1, both roots being the same. Thus, the homogeneous part of the solution will involve terms like C1ex and C2xex because of the repeated root.
To find a particular solution to the inhomogeneous equation, we can try a solution of the form Aex since the right-hand side is 6ex. Plugging this form into the differential equation gives us A = 6.
Therefore, the general solution to the differential equation is y = 6ex + C1ex + C2xex. Rearranging terms, we get y = (6 + C1)ex + C2xex. This implies that the correct answer is D) y = 6ex + C1ex + C2xex, since other variations with negative exponents or different powers of ex do not match the form we derived.